Microscopic diagonal entropy and its connection to basic thermodynamic relations
Files
Accepted manuscript
Date
2011-02-01
Authors
Polkovnikov, Anatoli
Version
Accepted manuscript
OA Version
Citation
Anatoli Polkovnikov. 2011. "Microscopic diagonal entropy and its connection to basic thermodynamic relations." ANNALS OF PHYSICS, Volume 326, Issue 2, pp. 486 - 499 (14). https://doi.org/10.1016/j.aop.2010.08.004
Abstract
We define a diagonal entropy (d-entropy) for an arbitrary Hamiltonian system as Sd = − 𝛴𝑛 𝘗𝑛𝑛 l𝑛 𝘗𝘯𝘯 with the sum taken over the basis of instantaneous energy states. In equilibrium this entropy coincides with the conventional von Neumann entropy Sn = −Trρ ln ρ. However, in contrast to Sn, the d-entropy is not conserved in time in closed Hamiltonian systems. If the system is initially in stationary state then in accord with the second law of thermodynamics the d-entropy can only increase or stay the same. We also show that the d-entropy can be expressed through the energy distribution function and thus it is measurable, at least in principle. Under very generic assumptions of the locality of the Hamiltonian and non-integrability the d-entropy becomes a unique function of the average energy in large systems and automatically satisfies the fundamental thermodynamic relation. This relation reduces to the first law of thermodynamics for quasi-static processes. The d-entropy is also automatically conserved for adiabatic processes. We illustrate our results with explicit examples and show that Sd behaves consistently with expectations from thermodynamics.